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# Did You Find Out? Get the Answer to the Riddle of E-News Issue No. 03, June 2016

2016-06-07

How often does Michael have to weigh the bags of apples?

Every weekend, farmer Michael offers his freshly grown produce at the weekly farmer's market. This weekend he has a lot of apples to offer for sale. He wants to sell them to the market visitors in pre-packaged bags. For this purpose, he weighs six different sized bags, from one kilogram up to six kilograms. Then Michael labels the individual bags in such a way so that each bag only contains as many kilograms of apples as the label on the package shows. The bag labeled with "1 kg" holds exactly one kilogram of apples. The bag with the label "2 kg" includes two kilograms of apples and so on.

Farmer Michael labels the six pre-packaged bags.

With the six pre-packaged bags, farmer Michael drives to the weekly market. Arriving there, he suddenly starts to wonder if he labeled the bags correctly. Only equipped with his simple balance scales, he wants to check the weights.

How often does Michael have to weigh the apples in order to find out whether he properly labeled the bags, or if he made a mistake when he applied the labels?

## Solution

Farmer Michael has only to weigh the apples twice in order to find out whether the bags were labeled correctly.

1. Weighing the first time

On one side of the balance scales you put the bag with the description "6 kg," and on the other side the three bags displaying the information "1 kg," "2 kg" and "3 kg".

Now there are two options:

• The scale is not balanced. In this case, the bags were not labeled correctly. If they were, the scales would have to be in balance.
• The scale is in balance. In this case you can conclude the following:
1) The bag with the description "6 kg" also contains six kilograms and was therefore properly labeled.
2) The three bags on the left-sided pan are in total six kilograms. There is no other way to have on one scale pan three sacks and on the other scale pan only one bag, and have it to be in balance without the total sum of six kilograms. "6 kg" is the only sum you can achieve with three sacks.

2. Weighing the second time

The second weighing you put the two bags with the descriptions "6 kg" and "1 kg" on the left scale pan and the two bags with the details "5 kg" and "3 kg" on the right scale pan. Only for this combination, the right scale pan is heavier than the left one. In this case, the bags "6 kg" and "1 kg" as well as "5 kg"and "3 kg" were labeled correctly. If the right scale pan is lighter than the left scale, it proves that there must be a mistake in the labels on the bags. What cases are conceivable? See the table below.

Conceivable cases to weigh the different bags of apples.

Finally, it has to be proved that the information on the other two bags "2 kg" and "4 kg" is correct.

The bags "1 kg," "2 kg," and "3 kg" have already been weighed together. Since the bags "1 kg" and "3 kg" were proved as correct, you can determine that the bag with the description "2 kg" is also labeled correctly - due to the exclusion principle. Now, only the bag "4 kg" is left. As all others were verified to be correct, it must be labeled correctly.

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